The curvature of a contraction T in the Cowen-Douglas class is bounded above by the curvature of the backward shift operator. However, 
 in general, an operator satisfying the curvature inequality need not be contractive. In this talk we characterize a slightly smaller class of 
 contractions using a stronger form of the curvature inequality.  Along the way, we find conditions on the metric of the holomorphic Hermitian 
 vector bundle E corresponding to the operator T in the Cowen-Douglas class which ensures negative definiteness of the curvature function. We 
 obtain a generalization for commuting tuples of operators in the Cowen-Douglas class.In the second part we will discuss the construction of a 
 Hermitian holomorphic jet bundle $J_k(E)$ starting with a Hermitian holomorphic vector bundle $E$ over an open subset $\Omega$ of $\mathbb{C}$. 
 In the special case, when the the Hermitian holomorphic vector bundle $E$ is the pull-back of the tutological bundle on the Grassmannian of some 
 Hilbert space $\mathcal{H}$, we will derive a compact formula for the curvature of the vector bundle. Several properties of the curvature of the 
 Hermitian holomorphic jet bundle $J_k(E)$ is then derived using the curvature formula for the vector bundle $E$.